balls and urn
Consider an urn holding N balls which are either white or black. Whenever I take out a ball it is replaced randomly with either a white or black ball,
with fixed probability p for the replacement ball being white.
I am playing this game for quite a while (so that the start configuration no longer matters) and now I am taking out n white balls in a row (n < N).
What is the probability that the next ball is also white?
If p is (near) 1/2 then one could make these two arguments:
1) If we take out a sequence of n white balls it indicates that there are probably
more white balls in the urn than black balls (due to a random fluctuation in the replacement process), so the next ball is most likely also white: P(white) > P(black).
2) If we take out n white balls, the ratio of white to black necessarily decreases,
so it is more likely that the next is actually black: P(black) > P(white).
What do you think? And does it make a difference if we actually know p and N ?
added later: I have posted the solution now as a comment, but I have to warn you that this is fun only if you really try to find an answer yourself first.
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3 comments:
I go for 2)
Initially the best guess is that #black = #white = N/2
Now we take out e.g. 10 white and the best guess is that they will be replaced with 5 white and 5 black.
So now the best estimate is #black = N/2 + 5 while #white = N/2 - 10 + 5
Am I right?
>> Am I right?
I'll post my answer later.
solution: The probability is p and independent of N and n.
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